It's basically a spacetime that admits that admits Cauchy surfaces. There's a theorem which states that all such spacetimes can be assigned continuous "time functions" [itex]t:M \rightarrow \mathbb{R}[/itex] where [itex]t^{-1}(s)[/itex] gives a Cauchy surface for any s. Also, if each Cauchy surface has topology [itex]\Sigma[/itex], the manifold has topology [itex]\Sigma \times \mathbb{R}[/itex]. Wald's GR book goes into these things extensively.

One of the less technical definitions is that it is a space-like surface representing an "instant of time" in the universe, and has the property that the future state of the universe and the past state of the universe can both be predicted/retrodicted from the values of "conditions" on the Cauchy Surface alone. (Of course this arises from a classical, deterministic viewpoint, but then GR is a classical theory, not a quantum theory).

The more technical defintion (also in Wald, as was this less technical defintion which I paraphrased a bit) involves a lot of discusion of achronal sets and domains of dependency.

It's "hyperbolic" as in "hyperbolic partial differential equation".
Check out this article from the Living Reviews site:
http://relativity.livingreviews.org/Articles/lrr-1998-3/node2.html#SECTION00011000000000000000 [Broken]

It is a spacetime in which
(a) no signals can come back arbitrarily close to themselves
(b) in which for any two events a,b where b is in the future of a one has a compact set of events c to the future of a and in the past of b
For more info, see Hawking and Ellis.